Van Barneveld blistered through the match at the MVM Dome, producing a three-dart average of 107.97, landing five maximums and closing it out with a clinical 144 checkout in a vintage display.
"Today it just clicked," he said, reflecting on his resurgence.
"I feel more positive. I often do not manage to take my form from the practice room onto the stage, but today it worked and I am really happy with this win."
His reward is a meeting with Price, who kicked off his campaign with a 6-2 win over Richard Veenstra.
It was a solid, if unspectacular, performance from Price, who admitted he held something in reserve: "It was not spectacular, but I did enough and I have got plenty more in the tank.
"I am playing fantastic darts this year. I just need a little bit of luck, but there is no reason why I cannot go and lift this trophy again."
Stars advance in Budapest
Other big names also had strong showings as the tournament passed the second round in front of a packed crowd.
Michael van Gerwen, Luke Humphries, and Luke Littler all advanced, none dropping to early nerves.
Van Gerwen, defending his title, beat Matthew Dennant 6-2, while Danny Noppert edged out Rob Owen in a tight one, surviving a match dart and having missed several of his own in the penultimate leg.
Humphries had to weather a 99 average from Nathan Aspinall before pulling through, aided by big scoring combinations (68, 72, 130) in the closing stages.
Littler and Martin Schindler also impressed: Littler demolished Joe Cullen 6-1, hitting six of his eight double attempts, and Schindler ran off five straight legs to beat Ryan Joyce 6-2 on Joyce's birthday.
In another highlight, Josh Rock came through a high-quality battle with Ricardo Pietreczko, surviving finishes of 124, 126 and 161 to win.
Thibault Tricole also staged a comeback, overturning a 5-2 deficit against Stephen Bunting, while Rob Cross, averaging just over 102, beat Daryl Gurney 6-3 to keep his European Championship hopes alive.
James Wade showed off his sharp touch too, with 135 and 156 checkouts in successive legs. He will face Kim Huybrechts next after the Belgian held his nerve to beat Peter Wright in a last-leg decider. Chris Dobey and Cameron Menzies also advanced, both grinding through tight encounters.
Round three now looms in Sunday's afternoon session, with the quarter-finals, semis and final set to close out in the evening.
